Riemann sums are a bridge between adding many simple pieces and measuring a curved area exactly. They approximate the area under a curve by slicing an interval into thin subintervals and building rectangles over them. This matters because many real quantities, such as distance from velocity or accumulated charge from current, are found by adding continuously changing values.
The definite integral is the exact value reached when the rectangles become infinitely thin.
On an interval [a, b], the width of each equal subinterval is Δx = (b - a)/n, and a sample point x_i* is chosen in each subinterval. The sum Σ f(x_i*) Δx adds the rectangle areas, and different choices give left, right, midpoint, or other Riemann sums. As n increases, the approximation usually improves, and if the limit exists, it is written as ∫_a^b f(x) dx.
This sigma-to-integral transition is one of the central ideas of calculus.
Key Facts
- For n equal subintervals on [a, b], Δx = (b - a)/n.
- A Riemann sum has the form Σ_{i=1}^n f(x_i*) Δx.
- The definite integral is defined by ∫_a^b f(x) dx = lim_{n→∞} Σ_{i=1}^n f(x_i*) Δx, when the limit exists.
- If f(x) ≥ 0 on [a, b], then ∫_a^b f(x) dx represents the area under the curve and above the x-axis.
- If f(x) is below the x-axis, its integral contributes negative signed area.
- For continuous functions, left, right, and midpoint Riemann sums all approach the same definite integral as n → ∞.
Vocabulary
- Riemann sum
- A Riemann sum is an approximation of a definite integral made by adding the areas of rectangles over subintervals.
- Definite integral
- A definite integral is the limiting value of Riemann sums over a fixed interval [a, b].
- Partition
- A partition is a division of an interval into smaller subintervals.
- Sample point
- A sample point is the x-value chosen inside a subinterval to determine the height of a Riemann rectangle.
- Signed area
- Signed area counts regions above the x-axis as positive and regions below the x-axis as negative.
Common Mistakes to Avoid
- Forgetting the factor Δx, which is wrong because f(x_i*) gives only a rectangle height, not an area.
- Using n instead of Δx as the rectangle width, which is wrong because n is the number of rectangles while Δx is the actual width of each rectangle.
- Assuming every definite integral is ordinary area, which is wrong because integrals measure signed area when the graph goes below the x-axis.
- Stopping at a finite Riemann sum and calling it exact, which is wrong because the definite integral is the limit as the number of subintervals approaches infinity.
Practice Questions
- 1 Approximate ∫_0^4 x^2 dx using a right Riemann sum with n = 4 equal subintervals.
- 2 For f(x) = 3x + 1 on [0, 2], compute the left Riemann sum with n = 4 equal subintervals.
- 3 Explain why increasing n in a Riemann sum usually makes the rectangle approximation closer to the definite integral for a continuous function.